Coordinated linear beamforming in downlink multi-cell wireless networks

ABSTRACT

System and methods are disclosed for optimizing wireless communication for a plurality of mobile wireless devices. The system uses beamforming vectors or precoders having a structure optimal with respect to the weighted sum rate in a multi-cell orthogonal frequency division multiple access (OFDMA) downlink. A plurality of base stations communicate with the mobile devices and all base stations perform a distributed non-convex optimization exploiting the determined structure.

The present application claims priority to U.S. Provisional ApplicationSer. No. 61/058,295, filed Jun. 3, 2008, the content of which isincorporated by reference.

BACKGROUND

This application relates to linear beamforming for wireless devices incellular networks.

Theoretical studies on the wireless broadcast channel have shown thatsignificant throughput gains are possible when multiple antenna elementsare installed at the transmitter and advanced spatial signal processingis employed to serve multiple co-channel terminals as opposed to theconventional time- and frequency-division multiple access. Inparticular, for an isolated network with M transmit antennas, nhomogeneous users equipped with N receive antennas and block Rayleighfading, the capacity of the multi-antenna Gaussian broadcast channelscales as M log₂ log₂(nN) for increasingly large n, achieving both amultiplexing gain (M) and a multiuser diversity gain log₂ log₂(nN).

A known capacity achieving strategy for the multi-antenna Gaussianbroadcast channel involves a highly complex non-linear interferencepre-cancellation technique, known as dirty paper coding (DPC), which isusually infeasible in practice. Several suboptimal strategies have beeninvestigated to overcome this limitation. Among them, linear transmitbeamforming has attracted great interests since it can achieve a largefraction of the capacity at much lower cost and implementationcomplexity. In linear beamforming, each data stream is modulated by aspatial signature vector before transmission through multiple antennas.Careful selection of the signature vectors can mitigate (or eveneliminate) mutual interference among different streams. Surprisingly,random unitary beamforming is sufficient to achieve the same sum-ratescaling growth M log₂ log₂(nN) of DPC for increasingly large number ofusers. However, when the number of users is smaller, random beamformingmay perform poorly and more effective beamforming strategies have beenproposed to achieve improved performance. They include for examplelinear minimum mean square error (MMSE) and zero-forcing (ZF)beamforming. Other works, instead, make use of the uplink-downlinkduality and design the transmit linear filters so as to minimize thetotal transmit power under given constraints on the users' rates. Othershave designed transmit linear filters in order to maximize the sum-rate.

Many existing works have addressed the beamforming design problem byassuming that each base station communicates with its respectiveterminals independently: in such framework, inter-cell interference issimply regarded as additional background noise and the design of thebeamforming vectors is performed on a per-cell basis only. However,future wireless cellular networks will be interference-limited; hence,significant performance gains are possible if inter-cell interference ismitigated via coordinated processing across multiple cells. Ideally, ifboth data and channel state information of all users could be shared inreal-time, all base stations could act as a unique large array withdistributed antenna elements and could employ joint beamforming,scheduling and data encoding to simultaneously serve multiple co-channelusers. In practice, a much lower level of coordination appears to befeasible depending on the bandwidth of the backbone network connectingthe base stations. For example, it may be reasonable to assume that eachuser is served by only one base station: in this case, the set ofdownlink beam-vectors can be optimized based on the inter-cell channelqualities. Also, complexity of the network infrastructure andsynchronization requirements may limit the number of coordinating basestations: in this case, coordination can be performed on a per-clusterbasis.

SUMMARY

System and methods are disclosed for optimizing wireless communicationfor a plurality of mobile wireless devices. The system uses linearbeamforming vectors having a structure optimal with respect to theweighted sum rate in a multi-cell orthogonal frequency division multipleaccess (OFDMA) downlink, comprising a plurality of mobile devices; and aplurality of base stations communicating with the mobile devices, allbase stations performing a distributed a non-convex optimizationexploiting the determined structure.

In one aspect, the system optimizes wireless communication for aplurality of mobile wireless devices by determining a structure fordownlink beamforming vectors optimizing the weighted sum-rate acrossmultiple orthogonal resource slots in the time, frequency or codedomain; and distributing a non-convex optimization exploiting thedetermined structure over a plurality of base stations.

Implementations of the above aspect may include one or more of thefollowing. The non-convex optimization is responsive to a set of channelestimates; user priorities; or maximum transmit power available at eachbase-station. The non-convex optimization utilizes Karush-Kuhn-Tucker(KKT) equations. The distribution over the base stations entails a smallnumber of channel estimates at each base station and limited messagepassing among adjacent base stations. The system can obtain channelestimates. The system can also obtain predicted channel estimatesresponsive to an average scheduling delay parameter. A multiple-inputsingle-output (MISO) system can be served wherein each user is equippedwith a single receive antenna and receives a single data stream from itsserving base station via space-division multiple-access (SDMA). Thesystem can determine the optimal structure of the beamforming vectorsoptimizing the weighted sum-rate in a MISO system. A leakage matrix canbe obtained, and in one implementation a leakage matrix L_(m,k)(n) canbe determined as:

${L_{m,k}(n)} = {{\sum\limits_{j = 1}^{M}{\sum\limits_{\substack{u \in {B_{j}{(n)}} \\ {({j,u})} \neq {({m,k})}}}^{\;}{{P_{j,u}(n)}{G_{m,u}(n)}}}} \in C^{P}}$where${P_{j,u}(n)} = \frac{{\alpha_{u}(n)}S\; I\; N\;{R_{j,u}(n)}}{1 + {\sum\limits_{l = 1}^{M}{\sum\limits_{s \in {B_{l}{(n)}}}{{w_{l,s}^{H}(n)}{G_{l,u}(n)}{w_{l,s}(n)}}}}}$

-   -   B_(m)(n) is the set of users served by base station m on slot n;    -   N is the total number of orthogonal resource slots;    -   M is the number of coordinated base stations;    -   P is the number of transmit antennas at each base station;    -   w_(m,k)(n)εC^(P) denotes the P-dimensional beam vector used by        the base station m to serve user kεB_(m)(n) on slot n;    -   W={w_(m,k)(n), kεB_(m)(n), m=1, . . . , M, n=1, . . . N}        indicates the collection of all beam vectors;    -   α_(k)(n)>0 is the priority assigned by the scheduler to user k        on slot n;    -   P_(m,max) is the maximum transmit power of base station m;    -   SINR_(m,k)(n) is the signal-to-interference-plus-noise ratio for        user k served by base station m on slot n and is given by

${S\; I\; N\;{R_{m,k}(n)}} = \frac{{w_{m,k}^{H}(n)}{G_{m,k}(n)}{w_{m,k}(n)}}{1 + {\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{{w_{j,u}^{H}(n)}{G_{j,k}(n)}{w_{j,u}(n)}}}}}$

-   -    with G_(m,k)(n)=h_(m,k)(n)h_(m,k) ^(H)(n)εC^(P×P).    -   h_(m,k)(n)εC^(P) is the P-dimensional channel vector between        base station j and user k and slot n (which includes small-scale        fading, large scale fading and path attenuation) normalized by        the standard deviation of the received noise.        The system can also determine

${{i_{m,k}(n)} = {\sum\limits_{l = 1}^{M}{\sum\limits_{\underset{{({l,s})} \neq {({m,k})}}{s \in {B_{l}{(n)}}}}{{w_{l,s}^{H}(n)}{G_{l,k}(n)}{w_{l,s}(n)}}}}},{k \in {B_{m}(n)}},{m = 1},\ldots\mspace{14mu},M,{n = 1},{\ldots\mspace{14mu}{N.}}$The determination of λ₁, . . . , λ_(m) and {{circumflex over(β)}_(m,k)(n;λ_(m)), kεB_(m)(n), n=1, . . . N} is:

${{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}}^{2} = \frac{\left( {{{\alpha_{k}(n)}{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}} - {{\hat{i}}_{m,k}(n)} - 1} \right)^{+}}{{{{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}$and${\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\underset{\underset{f_{m}{({n;\lambda_{m}})}}{︸}}{{{{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}}} = P_{m,\max}$where {circumflex over (T)}_(m,k)=(n;λ_(m))={circumflex over(L)}_(m,k)(n;λ_(m))+(λ_(m) ln 2)I_(P), (•)^(†) indicates thepseudo-inverse and x⁺=max{x,0} and where {î_(m,k)(n),{circumflex over(L)}_(m,k)(n)} denotes most recent values of {i_(m,k)(n),L_(m,k)(n)}.The system can update beam vectors asw _(m,k)(n)={circumflex over (β)}_(m,k)(n;λ _(m)){circumflex over (T)}_(m,k) ^(†)(n;λ _(m))h _(m,k)(n).The system can select an initial feasible set of beam vectors. Theinitial feasible set can be selected after splitting power across theavailable slots. The system can perform channel-matched beamforming toselect the initial feasible set of beam vectors. The system can also doin-cell zero-forcing beamforming to select the initial feasible set ofbeam vectors. Maximum signal-to-leakage-plus-noise ratio (MSLNR)beamforming can be used to select the initial feasible set of beamvectors. An equivalent MISO system can be provided wherein each mobileuser has multiple-receive antennas and performs rank-one receivebeamforming before detection. A multiple-input multiple-output (MIMO)system can be provided wherein each user is equipped with multiplereceive antennas and receives multiple data streams from a serving basestation via linear precoding. The optimal structure of the linearprecoders in a MIMO system can be determined. A leakage matrix can becomputed such as a leakage matrix L_(m,k)(n) where:

${{L_{m,k}( n)} = {\sum\limits_{j = 1}^{M}{\sum\limits_{\substack{u \in {B_{j}{(n)}} \\ {({j,u})} \neq {({m,k})}}}^{\;}{\frac{\alpha_{u}(n)}{\ln(2)}{H_{m,u}^{H}( n)}\left( {{R_{j,u}(n)}^{- 1} - \left( {{R_{j,u}(n)} + {{H_{j,u}(n)}{Q_{j,u}(n)}{H_{j,u}^{H}(n)}}} \right)^{- 1}} \right){H_{m,u}( n)}}}}},\mspace{79mu}{where}$$\mspace{79mu}{{R_{m,k}(n)} = {{\sum\limits_{j = 1}^{M}{\sum\limits_{\substack{u \in {B_{j}{(n)}} \\ {({j,u})} \neq {({m,k})}}}{{H_{j,k}(n)}{Q_{j,u}(n)}{H_{j,k}^{H}(n)}}}} + {I_{N_{k}}.}}}$

-   -   B_(m)(n) is the set of users served by base station m on slot n;    -   N is the total number of orthogonal resource slots;    -   M is the number of coordinated base stations;    -   P is the number of transmit antennas at each base station;    -   N_(k) is the number of receive antennas of user k;    -   Q_(j,u)(n)=W_(j,u)(n)W_(j,u) ^(H)(n)εC^(P×P) is a covariance        matrix to be optimized;    -   W_(m,k)(n)εC^(P×D) ^(m,k) ^((n)), and 1≦D_(m,k)(n)≦P indicates        the number of data stream delivered to user kεB_(m)(n) on slot n        by base station m.;    -   Q={Q_(m,k)(n)ε Q, kεB_(m)(n), m=1, . . . , M, n=1, . . . N}        indicates the collection of the positive-semidefinite covariance        matrices;    -   α_(k)(n)>0 is the priority assigned by the scheduler to user k        on slot n;    -   P_(m,max) is the maximum transmit power of base station m; and    -   H_(j,k)(n)εC^(N) ^(k) ^(×P) is the channel matrix between base        station j and user k and slot n (which includes small-scale        fading, large scale fading and path attenuation) normalized by        the standard deviation of the received noise.        The system can solve a following convex optimization problem at        each base-station m, for a given {L_(m,k)(n)} and {R_(m,k)(n)}:

${\underset{\{{Q_{m,k}{(n)}}\}}{argmax}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\left\lbrack {{{\alpha_{k}(n)}\log_{2}{{I_{N_{k}} + {{H_{m,k}(n)}{Q_{m,k}(n)}{H_{m,k}^{H}(n)}{R_{m,k}(n)}^{- 1}}}}} - {{tr}\left( {{L_{m,k}(n)}{Q_{m,k}(n)}} \right)}} \right\rbrack}}},{{s.t.\mspace{14mu}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(m)}}}{{tr}\left( {Q_{m,k}(n)} \right)}}}} \leq P_{m,\max}},{{Q_{m,k}(n)} \in \overset{\_}{Q}},{\forall{k \in {B_{m}(n)}}},{\forall n}$The system can select an initial feasible set of positive semi-definitecovariance matrices after splitting power across available slots.Maximum Signal-to-Leakage-plus-Noise Ratio (MSLNR) precoding can be usedto select the initial feasible set of positive semi-definite covariancematrices.

Advantages of the above systems and methods may include one or more ofthe following. By explicitly accounting for the inter-cell interferencein the design of the beam vectors, the invention leads to substantialimprovements in the system spectral efficiency (throughput).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a downlink multiuser system with P transmit-antennas at thebase station communicating with multiple mobile stations or devices.

FIG. 2 shows an exemplary multi-cell setup.

FIG. 3 shows an exemplary beamforming design process which utilizes aderived beamforming structure and solves Karush-Kuhn-Tucker (KKT)equations for the non-convex optimization problem.

DESCRIPTION OF FIGURES

FIG. 1 shows a downlink multiuser system, with P transmit-antennas atthe base station BS and 1 receive antenna at each user. A transmitter110 transmits from P transmitting antennas 111.1-111.P over a fadingchannel to multiple users. Each user's receive antenna is coupled to areceiver. At each user a channel estimator provides an estimate of thechannel corresponding to the propagation between the transmit antennasand the receive antenna, to the respective receiver. The channelestimate is also quantized and provided to the transmitter 110 via afeedback channel.

FIG. 2 shows an exemplary multi-cell setup that handles co-channelinterference mitigation in the downlink of a cellular system. In FIG. 2,all base stations are equipped with multiple transmit antennas and themobiles are equipped with a single or multiple receive antenna each.Every base station employs linear beamforming or linear precoding toserve each one of its scheduled users or mobile devices on any resourceslot and each scheduled user is served by only one base station. Theassociation of each user to a particular base station (its serving basestation) is pre-determined. The base station of each cell serves itsusers through linear beamforming or linear precoding. Each user receivesuseful (desired) signal from its serving base station (depicted usingthe solid line). Each user also receives interference (depicted as thedashed lines) from its serving base station as well as adjacent basestations. The interference includes all signals transmitted in the sameresource slot (i.e., time/frequency/code slot) as the desired signal butwhich are intended for the other users. The system designs a set of beamvectors for a given set of channel estimates, which maximizes theweighted system sum-rate over a cluster of coordinated cells subject toper-base-station power constraints.

Embodiment I:MISO System

First, a multiple-input single-output (MISO) embodiment is discussed.The MISO system includes a downlink cellular network with universalfrequency reuse where a cluster of M coordinated base stationssimultaneously transmit on N orthogonal (in the time or frequency orcode domain) resource slots during each scheduling interval. Each basestation is equipped with P antennas and space-division multiple-access(SDMA) is employed to serve multiple mobiles (each having one receiveantenna) on a resource slot. Each user is served by only one basestation and coordinated base stations only exchange channel qualitymeasurements and this scenario is referred to as a partially-connectedcluster (PCC). Let B_(m)(n) be the set of terminals scheduled by basestation m on slot n, for m=1, . . . , M and n=1, . . . , N. Forsimplicity and without loss of generality, |B_(m)(n)|=Q ∀ m, n and B_(m)₁ (n₁)∩B_(m) ₂ (n₂)=Ø for m₁≠m₂.

With linear preceding, the signal transmitted by base station m on slotn is expressed as

${{x_{m}(n)} = {{\sum\limits_{k \in {B_{m}{(n)}}}{{w_{m,k}(n)}{b_{m,k}(n)}}} \in C^{P}}},$where b_(m,k)(n) is the complex symbol transmitted by base station m onslot n to user kεB_(m)(n) using the beamforming vector w_(m,k)(n)εC^(P).One system uses the assumption that E[|b_(m,k)(n)|²]=1, E[b_(m) ₁ _(,k)₁ (n₁)b_(m) ₂ _(,k) ₂ (n₂)^(H)]=0 for (n₁, m₁, k₁)≠(n₂, m₂, k₂), and

${{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{w_{m,k}}^{2}}} \leq P_{m,\max}},$where P_(m,max) is the maximum transmit power of base station m;

Let {tilde over (h)}_(j,k)(n)εC^(P) be the channel matrix between basestation j and user k on slot n, which includes small-scale fading,large-scale fading and path attenuation. The signal received by userkεB_(m) (n) on slot n is given by

${{{\overset{\sim}{y}}_{k}(n)} = {{\sum\limits_{j = 1}^{M}{{{\overset{\sim}{h}}_{j,k}^{H}(n)}{x_{j}(n)}}} + {{\overset{\sim}{z}}_{k}(n)}}},$where {tilde over (z)}_(k)(n) is the additive circularly-symmetricGaussian noise with variance σ_(k) ²(n): here, different noise levelsaccount for different co-channel interference and different noisefigures of the receivers. To simplify notation, the received signal isnormalized by the noise standard deviation. Let y_(k)(n)={tilde over(y)}_(k)(n)/σ_(k)(n), z_(k)(n)={tilde over (z)}_(k)(n)/σ_(k)(n) andh_(k)(n)={tilde over (h)}_(k)(n)/σ_(k)(n), the following is obtained:

${y_{k}(n)} = {\underset{\underset{{useful}\mspace{14mu}{signal}}{︸}}{{h_{m,k}^{H}(n)}{w_{m,k}(n)}{b_{m,k}(n)}} + \underset{\underset{{co}\text{-}{channel}\mspace{14mu}{interference}}{︸}}{\sum\limits_{j = 1}^{M}{\underset{{({j,u})} \neq {({m,k})}}{\sum\limits_{u \in {B_{j}{(n)}}}^{\;}}{{h_{j,k}^{H}(n)}{w_{j,u}(n)}{b_{j,u}(n)}}}} + {\underset{\underset{noise}{︸}}{z_{k}(n)}.}}$The objective function to be maximized is the (instantaneous) weightedsystem sum-rate subject to per-base-station power constraints. AssumingGaussian inputs and single-user detection at each mobile, the problem tobe solved is as follows:

$\begin{matrix}{{\arg\;{\max\limits_{W}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{\sum\limits_{k \in {B_{m}{(n)}}}{{\alpha_{k}(n)}{\log_{2}\left( {1 + {S\; I\; N\;{R_{m,k}(n)}}} \right)}}}}}}}{{s.t.\mspace{14mu}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{w_{m,k}(n)}}^{2}}}} \leq P_{m,\max}}{{m = 1},\ldots\mspace{14mu},M}} & \left\lbrack {{problem}\mspace{14mu} 1} \right\rbrack\end{matrix}$where

-   -   W={w_(m,k)(n), kεB_(m)(n), m=1, . . . , M, n=1, . . . N}        indicates the collection of all beam vectors.    -   α_(k)(n)>0 is the priority assigned by the scheduler to user k        on slot n; if α_(k)(n)=1/(NM), the objective function becomes        the network sum-rate (measured in bits/channel-use/slot/cell);        more generally, the scheduler may adjust the coefficients        {α_(k)(n)} over time to maintain proportional fairness among        terminals.    -   SINR_(m,k)(n) is the signal-to-interference-plus-noise ratio for        user k served by base station m on slot n and is given by

${{S\; I\; N\;{R_{m,k}(n)}} = \frac{{w_{m,k}^{H}(n)}{G_{m,k}(n)}{w_{m,k}(n)}}{1 + {\sum\limits_{j = 1}^{M}{\underset{{({j,u})} \neq {({m,k})}}{\sum\limits_{\mspace{14mu}{u \in {B_{j}{(n)}}}}}{{w_{j,u}^{H}(n)}{G_{j,k}(n)}{w_{j,u}(n)}}}}}},$

-   -   with G_(m,k)(n)=h_(m,k)(n)h_(m,k) ^(H)(n)εC^(P×P).

The system first applies Lagrangian theory to derive the structure ofthe optimal beam-vectors; then, a process iteratively solves the setKarush-Kuhn-Tucker (KKT) conditions corresponding to the non-convexprimal problem. A distributed implementation of the proposed algorithmis illustrated. Also, several approaches to choose the initialbeam-vectors can be followed. Finally, an extension of the proposedalgorithm to an equivalent MISO system is discussed.

FIG. 3 shows an exemplary process which utilizes the derived beamformingstructure and solves the Karush-Kuhn-Tucker (KKT) equations for thenon-convex optimization problem.

In 200, the system is initialized by obtaining the estimates of allchannels, the maximum transmit power of each base station and thepriorities of the users. The system selects some initial values for thebeam vectors {w_(m,k)(n), kεB_(m)(n), m=1, . . . , M, n=1, . . . . N}.Moreover, the values of the maximum number of iterations for the innerand outer loops L_(in,max) and L_(out,max), respectively, are selected.Further, the system sets the counter l_(out)=0.

Next, in 201, the system sets l_(in) to zero and computes the leakagematrices {L_(m,k)(n), kεB_(m)(n), m=1, . . . , M, n=1, . . . N}, where

${{L_{m,k}(n)} = {\sum\limits_{j = 1}^{M}{\underset{{({j,u})} \neq {({m,k})}}{\sum\limits_{u \in {B_{j}{(n)}}}}{{P_{j,u}(n)}{G_{m,u}(n)}}}}},{and}$${P_{j,u}(n)} = {\frac{{\alpha_{u}(n)}S\; I\; N\;{R_{j,u}(n)}}{1 + {\sum\limits_{l = 1}^{M}{\sum\limits_{s \in {B_{l}{(n)}}}{{w_{l,s}^{H}(n)}{G_{l,u}(n)}{w_{l,s}(n)}}}}}.}$

In 202, the system determines the scalars

${{i_{m,k}(n)} = {\sum\limits_{l = 1}^{M}{\underset{{({l,s})} \neq {({m,k})}}{\sum\limits_{s \in {B_{l}{(n)}}}^{\;}}{{w_{l,s}^{H}(n)}{G_{l,k}(n)}{w_{l,s}(n)}}}}},{k \in {B_{m}(n)}},{m = 1},\ldots\mspace{14mu},M,{n = 1},{\ldots\mspace{14mu}{N.}}$

In 203, for each base station m, the system determines λ_(m) and

{β̂_(m, k)(n; λ_(m)), k ∈ B_(m)(n), n = 1, …  N}   using${{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}}^{2} = \frac{\left( {{{\alpha_{k}(n)}{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}} - {{\hat{i}}_{m,k}(n)} - 1} \right)^{+}}{{{{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}$and${{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\underset{\underset{f_{m}{({n;\lambda_{m}})}}{︸}}{{{{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}}} = P_{m,\max}},$where x⁺=max{x,0}, {circumflex over (T)}_(m,k)(n;λ_(m))={circumflex over(L)}_(m,k)(n;λ_(m))+(λ_(m) ln 2)I_(p) and (•)^(†) indicates thepseudo-inverse. Notice that {î_(m,k)(n),{circumflex over (L)}_(m,k)(n)}denote the recent-most values of {i_(m,k)(n),L_(m,k)(n)} (i.e., thevalues most-recently updated). Finally, the system updates the beamvectors {w_(m,k)(n)} usingw _(m,k)(n)={circumflex over (β)}_(m,k)(n;λ _(m)){circumflex over (T)}_(m,k) ^(†)(n;λ _(m))h _(m,k)(n).

In 204, l_(in) is incremented. In 205 the system checks if the choice ofbeams has converged or if l_(in)=L_(in,max). If either of these twoconditions is true the process proceeds to 206 else the process loopsback to 202. In 206, the process increments l_(out).

In 207, the system checks if the choice of beams has converged or ifl_(out)=L_(out,max). If either of these two conditions is true thesystem jumps to 208 else loops back to 201. In 208, the latest choice ofbeams is provided as an output and the process ends.

The details of the iterative process of FIG. 3 to solve the KKTconditions are discussed next. It can be shown that the set of KKTconditions for the [problem 1] are given by

$\begin{matrix}{\quad\left\{ \begin{matrix}{{{\begin{pmatrix}{{L_{m,k}(n)} +} \\{\lambda_{m}\ln\; 2I_{P}}\end{pmatrix}{w_{m,k}(n)}} = \frac{{\alpha_{k}(n)}{G_{m,k}(n)}{w_{m,k}(n)}}{1 + {{w_{m,k}^{H}(n)}{G_{m,k}(n)}{w_{m,k}(n)}} + {i_{m,k}(n)}}}\mspace{79mu}{{k \in {B_{m}(n)}},{m = 1},\ldots\mspace{14mu},M,{n = 1},{\ldots\mspace{14mu}{N.}}}} \\{{{{\lambda_{m}\left\lbrack {P_{m,\max} - {\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{w_{m,k}(n)}}^{2}}}} \right\rbrack} = 0},\mspace{14mu}{m = 1},\ldots\mspace{14mu},M}\;} \\{{{{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{w_{m,k}(n)}}^{2}}} \leq P_{m,\max}},\mspace{14mu}{m = 1},\ldots\mspace{14mu},M}\;}\end{matrix} \right.} & \left\lbrack {{KKT}\mspace{14mu} 1} \right\rbrack\end{matrix}$

Notice that, if w_(m,k)(n)≠0 and λ_(m)≠0, then h_(m,k)^(H)(n)w_(m,k)(n)≠0 and h_(m,k)(n)∝(L_(m,k)(n)+λ_(m) ln2I_(P))w_(m,k)(n). Therefore, a non-zero beam-vector which satisfies theabove KKT conditions must be of the form w_(m,k)(n)∝(L_(m,k)(n)+λ_(m) ln2I_(P))⁻¹h_(m,k)(n). On the other hand, if w_(m,k)(n) ≠0, λ_(m)=0 isfeasible only if one of the following two conditions holds:

C1: h_(m,k) ^(H)(n)w_(m,k)(n)=0 and L_(m,k)(n)w_(m,k)(n)=0;

C2: h_(m,k)(n)εrange{L_(m,k)(n)}.

If C1 is satisfied, then user kεB_(m)(n) receives no information fromthe serving base station. On the other hand, if C2 holds and λ_(m)=0, anon-zero beam-vector which satisfies the KKT conditions must be of theform w_(m,k)(n)∝L_(m,k) ^(†)(n)h_(m,k)(n). LetT_(m,k)(n;λ_(m))=L_(m,k)(n)+(λ_(m) ln 2)I_(P), then the optimal non-zerobeam-vectors are of the form w_(m,k)(n)=β_(m,k)(n;λ_(m))T_(m,k)^(†)(n;λ_(m))h_(m,k)(n), kεB_(m)(n), m=M, n=1, . . . N, where λ_(m)≧0and β_(m,k)(n;λ_(m))>0 are constants to be determined. Also, λ_(m)=0 isfeasible only if h_(m,k)(n)εrange{L_(m,k)(n)} for k (B_(m)(n) and n=1, .. . N.

If w_(m,k)(n)≠0, the following is obtained:α_(k)(n)h _(m,k) ^(H)(n)T _(m,k) ^(†)(n;λ _(m))h _(m,k)(n)=1+i_(m,k)(n)+|β_(m,k)(n;λ _(m))|² |h _(m,k) ^(H)(n)T _(m,k) ^(†)(n;λ _(m))h_(m,k)(n)|² , kεB _(m)(n), m=1, . . . , M, n=1, . . . , N,and

${{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{{\beta_{m,k}\left( {n;\lambda_{m}} \right)}{T_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}} \leq P_{m,\max}},{m = 1},\ldots\mspace{14mu},{M.}$The procedure to solve the KKT system is as follows. If some previousfeasible vectors {Ŵ_(m,k)(n)} are given, then {î_(m,k)(n)} and{{circumflex over (L)}_(m,k)(n)} can be computed. The new values of{{circumflex over (β)}_(m,k) (n;λ_(m))} can now be computed as

${{{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}}^{2} = \frac{\left( {{{\alpha_{k}(n)}{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}} - {{\hat{i}}_{m,k}(n)} - 1} \right)^{+}}{{{{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}},$for kεB_(m)(n), m=1, . . . , M and n=1, . . . , N. The new value ofλ_(m) can be computed as the solution to

${\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\underset{\underset{f_{m}{({n;\lambda_{m}})}}{︸}}{{{{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}}} = {P_{m,\max}.}$In order to solve the above equation, notice that ƒ_(m)(n;λ_(m)) is amonotonic decreasing function of λ_(m)>0, for m=1, . . . , M. Moreover,the optimal set of dual variables λ₁, . . . , λ_(M) must satisfy thefollowing inequalities:

${0 \leq {\lambda_{m}\ln\; 2} \leq {\max\limits_{n \in {\{{1,\ldots\;,N}\}}}{\max\limits_{k \in {B_{m}{(n)}}}{{\alpha_{k}(n)}{{h_{m,k}(n)}}^{2}}}}},{m = 1},\ldots\mspace{14mu},{M.}$

Hence, the search for λ_(m) can be efficiently solved viaone-dimensional bisection. If no positive solution exists, then λ_(m) isset to zero: in this case, base station m does not use all availablepower for transmission. After computing the new values of {λ_(m)} and{β_(m,k)(n;λ_(m))}, the beam-vectors are updated and the process isiterated until convergence. This process is called the IterativeCoordinated Beam-Forming (ICBF) process. One exemplary pseudo code forthe ICBF is as follows:

 1: Initialize L_(in,max), L_(out,max), {w_(m,k)(n),k ∈ B_(m)(n), m =1,...,M,n =    1,...,N}  2: l_(out) = 0  3: repeat  4: Compute{L_(m,k)(n),k ∈ B_(m)(n),m = 1,...,M,n = 1,...,N}  5: l_(in) = 0 6: repeat  7:  Compute {i_(m,k)(n),k ∈ B_(m)(n),m = 1,...,M,n =1,...,N}  8:  for m = 1 to M do  9:   Compute λ_(m) and {{circumflexover (β)}_(m,k)(n;λ_(m)),k ∈ B_(m)(n),n = 1,...,N} 10:    Update{w_(m,k)(n),k ∈ B_(m)(n),n = 1,...,N} 11:   end for 12:   l_(in) =l_(in) + 1 13:  until convergence or l_(in) = L_(in,max) 14:  l_(out) =l_(out) + 1 15: until convergence or l_(out) = L_(out,max)

Next, a distributed implementation of the ICBF process is discussedwhich requires limited channel state information at each base stationand some inter base station message passing. The co-channel interferencereceived at terminal kεB_(m)(n) on slot n can be expanded as

${{i_{m,k}(n)} = {\underset{\underset{i_{m,k}^{i\; n}{(n)}}{︸}}{\sum\limits_{u \in \underset{u \neq k}{B_{m}{(n)}}}{{w_{m,u}^{H}(n)}{G_{m,k}(n)}{w_{m,u}(n)}}} + \underset{\underset{i_{m,k}^{out}{(n)}}{︸}}{\underset{j \neq m}{\sum\limits_{j = 1}^{M}}\underset{\underset{i_{m,k}^{{out},j}{(n)}}{︸}}{\sum\limits_{u \in {B_{j}{(n)}}}{{w_{j,u}^{H}(n)}{G_{j,k}(n)}{w_{j,u}(n)}}}}}},$where the terms i_(m,k) ^(in)(n) and i_(m,k) ^(out)(n) account forin-cell and out-cell interference, respectively. Similarly, the leakagematrix of terminal kεB_(m)(n) on slot n becomes

${{L_{m,k}(n)} = {\underset{\underset{L_{m,k}^{i\; n}{(n)}}{︸}}{\sum\limits_{u \in \underset{u \neq k}{B_{m}{(n)}}}^{\;}{{P_{m,u}(n)}{G_{m,u}(n)}}} + \underset{\underset{L_{m}^{out}{(n)}}{︸}}{\underset{j \neq m}{\sum\limits_{j = 1}^{M}}\underset{\underset{L_{m}^{{out},j}{(n)}}{︸}}{\sum\limits_{u \in {B_{j}{(n)}}}{{P_{j,u}(n)}{G_{m,u}(n)}}}}}},$where the terms L_(m,k) ^(in)(n) and L_(m) ^(out)(n) account for in-celland out-cell leakages, respectively, and

${P_{m,u}(n)} = {\frac{{\alpha_{u}(n)}{w_{m,u}^{H}(n)}{G_{m,u}(n)}{w_{m,u}(n)}}{\left( {1 + {i_{m,u}(n)}} \right)\left( {1 + {i_{m,u}(n)} + {{w_{m,u}^{H}(n)}{G_{m,u}(n)}{w_{m,u}(n)}}} \right)}.}$

Base station m can compute {i_(m,k) ^(in)(n), kεB_(m)(n), n=1, . . . ,N} based only on the knowledge of the local forward channels{h_(m,k)(n), kεB_(m)(n), n=1, . . . , N} and of the local beam-vectorsW_(m)={w_(m,k)(n), kεB_(m) (n), n=1, . . . , N} computed at the previousiteration. Furthermore, assume that base station m has received at theend of the previous iteration the quantities {i_(m,k) ^(out,j)(n),kεB_(m)(n), n=1, . . . , N} and {L_(m) ^(out,j)(n), n=1, . . . , N} fromevery other coordinated base station j≠m. Then, the base station m canlocally update its beam-vectors W_(m) as discussed above.

Once the new beam vectors are computed, each base station m can firstupdate the quantities

${{i_{l,s}^{{out},m}(n)} = {\sum\limits_{u \in {B_{m}{(n)}}}{{w_{m,u}^{H}(n)}{G_{m,s}(n)}{w_{m,u}(n)}}}},{l \neq m},{s \in {B_{l}(n)}},$and pass them to the other coordinated base stations. Upon receiving{i_(m,k) ^(out,j)(n)} from all j≠m and updating P_(m,u)(n), base stationm can determine

${{L_{l}^{{out},m}(n)} = {\sum\limits_{u \in {B_{m}{(n)}}}{P_{m,u}{G(n)}_{l,u}(n)}}},{l \neq m},$which are then passed to the other coordinated base stations for thenext iteration. The base station m needs the knowledge of the forwardchannels to the other coordinated cells and of the forward channels fromthe other coordinated cells.

Notice that a set of {i_(m,k)(n),L_(m,k)(n)} and beam-vectors W_(m) thatsatisfy [KKT 1], must also satisfy the KKT conditions of the following Moptimization problems (one for each base station) where{i_(m,k)(n),L_(m,k)(n)} are held fixed

${\max\limits_{W_{m}}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in_{m}{(n)}}{{\alpha_{k}(n)}{\log_{2}\left( {1 + \frac{\begin{matrix}{{w_{m,k}^{H}(n)}{G_{m,k}(n)}} \\{w_{m,k}(n)}\end{matrix}}{1 + {i_{m,k}(n)}}} \right)}}}}} - \frac{{w_{m,k}^{H}(n)}{L_{m,k}(n)}{w_{m,k}(n)}}{\ln\; 2}$$\mspace{79mu}{{s.t.\;{\sum\limits_{n = 1}^{N}\;{\sum\limits_{k \in {B_{m}{(n)}}}^{\;}{{w_{m,k}(n)}}^{2}}}} \leq P_{m,\max}}$For m=1, . . . , M, the above problem is equivalent to a second-ordercone program (SOCP) convex optimization problem. Consequently, at eachbase station m an optimal set of beam vectors W_(m) can also bedetermined by using a SOCP solver.

Next, the initialization of the ICBF process is discussed. Implementingthe ICBF process requires choosing an initial feasible set ofbeam-vectors. Three initial solutions can be used where the availablepower is equally splits across beams.

The first is called Channel-Matched (CM) Beamforming. Neglecting in-celland inter-cell interference, the system selects the initial beam vectorsmatched to the users' channel:

${{w_{m,k}(n)} = {\sqrt{\frac{P_{m,\max}}{NQ}}\frac{h_{m,k}(n)}{{h_{m,k}(n)}}}},$for kεB_(m)(n), m=1, . . . , M and n=1, . . . , N.

The second is called In-Cell Zero-Forcing (ICZF) Beamforming. If P≧Q,the initial set of beams can be designed so as to eliminate the in-cellinterference, i.e.,

${{w_{m,k}(n)} \propto {\left( {\sum\limits_{u \in {B_{m}{(n)}}}^{\;}{{h_{m,u}(n)}{h_{m,u}^{H}(n)}}} \right)^{\dagger}{h_{m,k}(n)}}},$for kεB_(m)(n), m=1, . . . , M and n=1, . . . , N. The proportionalityconstant is chosen to meet the power constraint. If P<Q, in-cellinterference cannot be forced to zero anymore. However, the aboveequation is still a wise beamformer choice since it maximizes the ratiobetween the useful signal and the in-cell leakage for any userkε=B_(m)(n).

The third alternative is the Maximum Signal-to-Leakage-plus-Noise Ratio(MSLNR) Beamforming. For any user kεB_(m)(n) the SLNR on slot n can bedefined as

${S\; L\; N\;{R_{m,k}(n)}} = {\frac{\overset{\overset{{useful}\mspace{14mu}{signal}\mspace{14mu}{for}\mspace{14mu}{user}\mspace{14mu} k\mspace{14mu}{on}\mspace{14mu}{slot}\mspace{14mu} n}{︷}}{{w_{m,k}^{H}(n)}{G_{m,k}(n)}{w_{m,k}(n)}}}{{\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}\underset{\underset{{leakage}\mspace{14mu}{to}\mspace{14mu}{user}\mspace{14mu} u\mspace{14mu}{on}\mspace{14mu}{slot}\mspace{14mu} n}{︸}}{{w_{m,k}^{H}(n)}{G_{m,u}(n)}{w_{m,k}(n)}}}} + 1}.}$w_(m,k)(n) is designed to maximize SLNR_(m,k)(n) subject to a normconstraint:

${{w_{m,k}(n)} = {\underset{x \in C^{P}}{argmax}\frac{x^{H}{G_{m,k}(n)}x}{x^{H}{D_{m,k}(n)}x}}},{{s.t.\mspace{14mu}{x}^{2}} = \frac{P_{m,\max}}{NQ}},{where}$${D_{m,k}(n)} = {{\sum\limits_{j = 1}^{M}{\underset{{({j,u})} \neq {({m,k})}}{\sum\limits_{u \in {B_{j}{(n)}}}^{\;}}{G_{m,u}(n)}}} + {I_{p}{\frac{NQ}{P_{m,\max}}.}}}$This is a standard problem whose solution is the generalized eigenvectorcorresponding to the maximum generalized eigenvalue of the matrix pair(G_(m,k))(n),D_(m,k)(n)).

The foregoing implementations of the ICBF process are referred to asICBF-CM or ICBF-ICZF or ICBF-MSLNR.

If the coordinated base stations share both user data and channel stateinformation, then they can act as a unique large array with distributedantenna elements and this case is the fully-connected cluster (FCC). AnFCC encompassing {tilde over (M)} base stations, each one equipped with{tilde over (P)} transmit antennas and where power pooling across basestations is possible, is mathematically equivalent to a PCC with M=1 andP={tilde over (M)}{tilde over (P)}. For such an FCC the optimal weightedsum rate is obtained using dirty-paper coding (DPC) and can bedetermined. Applying the ICBF algorithm (to the equivalent PCC model)the system can obtain a locally optimal solution.

The ICBF process can be generalized to an FCC architecture with powerpolling and a weighted sum-power constraint across base stations. Thismay be attractive when the coordinated base stations pay unequal pricesfor transmit power: in this latter case, a larger weight can be given toa base station which has a stricter transmit power limitation.

The ICBF process can be also generalized to an equivalent MISO systemwherein each mobile user has multiple-receive antennas and performsrank-one receive beamforming before detection. Details are given in thefollowing.

Let N_(k) be the number of receive antennas of user k. Then, the signalreceived by user kεB_(m)(n) on slot n is given by

${{{\overset{\sim}{r}}_{k}(n)} = {{{\sum\limits_{j = 1}^{M}{{{\overset{\sim}{H}}_{j,k}(n)}{x_{j}(n)}}} + {{\overset{\sim}{q}}_{k}(n)}} \in C^{N_{k}}}},$where {tilde over (H)}_(j,k)(n)εC^(N) ^(k) ^(×P) is the channel matrixbetween base station j and user k on slot n (which includes small-scalefading, large-scale fading and path attenuation), x_(j)(n)εC^(P) is thesignal transmitted by base station m on slot n and {tilde over(q)}_(k)(n)εC^(N) ^(k) is the additive circularly-symmetric Gaussiannoise with covariance matrix σ_(k) ²(n)I_(N) _(k) . Upon definingr_(k)(n)={tilde over (r)}_(k)(n)/σ_(k)(n), q_(k)(n)={tilde over(q)}_(k)(n)/σ_(k)(n) and H_(k)(n)={tilde over (H)}_(k)(n)/σ_(k)(n), thefollowing is obtained

${r_{k}(n)} = {\underset{\underset{{useful}\mspace{14mu}{signal}}{︸}}{{H_{m,k}(n)}{w_{m,k}(n)}{b_{m,k}(n)}} + {\quad{{\underset{\underset{{co}\text{-}{channel}\mspace{14mu}{interference}}{︸}}{\sum\limits_{j = 1}^{M}{\underset{{({j,u})} \neq {({m,k})}}{\sum\limits_{u \in {B_{j}{(n)}}}^{\;}}{{H_{m,k}(n)}{w_{j,u}(n)}{b_{j,u}(n)}}}} + \underset{\underset{noise}{︸}}{q_{k}(n)}} \in {C^{N_{k}}.}}}}$

Assume now that a receive beamforming vector u_(k)(n)εC^(N) ^(k) , withunit-norm, is employed by user k on slot n. The corresponding outputsignal y_(k)(n)=u_(k) ^(H)(n)r_(k)(n) is written as

${{y_{k}(n)} = {\underset{\underset{{useful}\mspace{14mu}{signal}}{︸}}{{h_{m,k}^{H}(n)}{w_{m,k}(n)}{b_{m,k}(n)}} + \underset{\underset{{co}\text{-}{channel}\mspace{14mu}{interference}}{︸}}{\sum\limits_{j = 1}^{M}{\underset{{({j,u})} \neq {({m,k})}}{\sum\limits_{u \in {B_{j}{(n)}}}}{{h_{j,k}^{H}(n)}{w_{j,u}(n)}{b_{j,u}(n)}}}} + \underset{\underset{noise}{︸}}{z_{k}(n)}}},$where h_(m,k) ^(H)(n)=u_(k) ^(H)(n)H_(m,k)(n)εC^(1×P), andz_(k)(n)=U_(k) ^(H)(n)q_(k)(n) C. This is an equivalent MISO signalmodel.

In the equivalent MISO system model, the transmit linear filters and thereceive linear filters can be iteratively designed as follows:

For any given choice of the receive linear filters {u_(k)(n),kεB_(m)(n), m=1, . . . , M, n=1, . . . N}, the transmit linear filters{w_(m,k)(n), kεB_(m)(n), m=1, . . . , M, n=1, . . . N} can be optimizedaccording to the ICBF algorithm described in FIG. 3.

Also, for a given choice of the transmit linear filters {w_(m,k)(n),kεB_(m)(n), m=1, . . . , M, n=1, . . . N}, the receive linear filters{u_(k)(n), kεB_(m)(n), m=, . . . , M, n=1, . . . N} can be chosen so asto maximize the weighted sum-rate. In particular, the optimal receivefilters must minimize the mean square estimation error and are given by:

${{u_{k}(n)} \propto {\left( {{\sum\limits_{j = 1}^{M}{\sum\limits_{u \in {B_{j}{(n)}}}{{H_{m,k}(n)}{w_{j,u}(n)}{w_{j,u}^{H}(n)}{H_{m,k}^{H}(n)}}}} + I_{N_{k}}} \right)^{- 1}{H_{m,k}(n)}{w_{m,k}(n)}}},$where u=k is not included in the summation and the proportionalityconstant is chosen to have ∥u_(k)(n)∥=1.Embodiment II: MIMO-System

Next, a MIMO embodiment is discussed. The MIMO system includes adownlink cellular network with universal frequency reuse where a clusterof M coordinated base stations simultaneously transmit on N orthogonal(in the time or frequency or code domain) resource slots during eachscheduling interval. Each base station is equipped with P antennas andSDMA is employed to serve multiple mobiles (each having multiple receiveantennas) on a resource slot. Let N_(k) be the number of receiveantennas of user k. Each user is served by only one base station andcoordinated base stations only exchange channel quality measurements andthis scenario is referred to as a partially-connected cluster (PCC). LetB_(m)(n) be the set of terminals scheduled by base station m on slot n,for m=1, . . . , M and n=1, . . . , N. For simplicity and without lossof generality, the assumption is that |B_(m)(n)|=Q ∀ m, n and B_(m) ₁(n₁)∩B_(m) ₂ (n₂)=Ø for m₁≠m₂.

With linear precoding, the signal transmitted by base station m on slotn is expressed as

${{x_{m}(n)} = {{\sum\limits_{k \in {B_{m}{(n)}}}{{W_{m,k}(n)}{b_{m,k}(n)}}} \in C^{P}}},$where b_(m,k)(n)εC^(D) ^(m,k) ^((n)) is the complex symbol vectortransmitted by base station m on slot n to user kεB_(m)(n) using theprecoder matrix W_(m,k)(n)εC^(P×D) ^(m,k) ^((n)), and 1≦D_(m,k)(n)≦Pindicates the number of data stream delivered to user kεB_(m)(n) on slotn by base station m. Also, the assumption is E[b_(m,k)(n)b_(m,k)^(H)(n)]=I_(D) _(m,k) _((n)), E[b_(m) ₁ _(,k) ₁ (n₁)b_(m) ₂ _(,k) ₂^(H)(n₂)]=0_(D) _(m,k) _((n)) for (n₁, m₁, k₁)≠(n₂, m₂, k₂), and

${{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {{W_{m,k}(n)}{W_{m,k}^{H}(n)}} \right)}}} \leq P_{m,\max}},$where tr(.) denotes the trace operation.

Let {tilde over (H)}_(j,k)(n)εC^(N) ^(k) ^(×P) be the channel matrixbetween base station j and user k on slot n, which includes small-scalefading, large-scale fading and path attenuation. The signal received byuser kεB_(m)(n) on slot n is given by

${{{\overset{\sim}{y}}_{k}(n)} = {{\sum\limits_{j = 1}^{M}{{{\overset{\sim}{H}}_{j,k}(n)}{x_{j}(n)}}} + {{\overset{\sim}{z}}_{k}(n)}}},$where {tilde over (z)}_(k)(n)εC^(N) ^(k) is the additivecircularly-symmetric Gaussian noise with covariance matrix σ_(k)²(n)I_(N) _(k) : here, different noise levels account for differentco-channel interference and different noise figures of the receivers. Tosimplify notation, the received signal is normalized by the noisestandard deviation. Upon defining y_(k)(n)={tilde over(y)}_(k)(n)/σ_(k)(n), z_(k)(n)={tilde over (z)}_(k)(n)/σ_(k)(n) andH_(k)(n)={tilde over (H)}_(k)(n)/σ_(k)(n), the following is derived:

${y_{k}(n)} = {\underset{\underset{{useful}\mspace{14mu}{signal}}{︸}}{{H_{m,k}(n)}{W_{m,k}(n)}b_{m,{k{(n)}}}} + \underset{\underset{{co}\text{-}{channel}\mspace{14mu}{interface}}{︸}}{\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{{H_{j,k}(n)}{W_{j,u}(n)}{b_{j,u}(n)}}}} + \;{z{\underset{\underset{noise}{︸}}{\,_{k}(n)}.}}}$

Moreover, let Q_(j,u)(n)=W_(j,u)(n)W_(j,u) ^(H)(n)εC^(P×P), so that thecovariance matrix of the co-channel interference plus noise seen by userkεB_(m)(n) on slot n, denoted by R_(m,k)(n), can be expressed as

${R_{m,k}(n)} = {{\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{{H_{j,k}(n)}{Q_{j,u}(n)}{H_{j,k}^{H}(n)}}}} + {I_{N_{k}}.}}$

Let Q denote the set of all P×P positive semi-definite matrices inC^(P×P). Let Q={Q_(m,k)(n)ε Q, kεB_(m)(n), m=1, . . . , M, n=1, . . . N}indicate the collection of the positive-semidefinite covariancematrices. Q is designed so as to maximize the (instantaneous) weightedsystem sum-rate. Assuming Gaussian inputs and single-user detection ateach mobile, the problem to be solved is as follows:

${\underset{Q}{argmax}{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{\sum\limits_{k \in {B_{M}{(n)}}}{{\alpha_{k}(n)}\log_{2}{{I_{N_{k}} + {{H_{m,k}(n)}{Q_{m,k}(n)}{H_{m,k}^{H}(n)}{R_{m,k}(n)}^{- 1}}}}}}}}},\mspace{79mu}{{s.t.\mspace{11mu}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {Q_{m,k}(n)} \right)}}}} \leq P_{m,\max}},{m = 1},\ldots\mspace{14mu},M,$where |.| denotes the determinant of its matrix argument, α_(k)(n)>0 isthe priority assigned by the scheduler to user k on slot n. Ifα_(k)(n)=1/(NM), the objective function becomes the network sum-rate(measured in bits/channel-use/slot/cell). More generally, the schedulermay adjust the coefficients {α_(k)(n)} over time to maintainproportional fairness among terminals.

Next, the first-order necessary conditions for the optimal solutions tothe above equations, namely the Karush-Kuhn-Tucker (KKT) conditions arediscussed. Consider the Lagrangian of the optimization problem dualizedwith respect to the sum-power constraints and positivity constraints:

${{\Lambda\left( {Q,\lambda} \right)} = {{\sum\limits_{n = 1}^{N}{\sum\limits_{m = 1}^{M}{\sum\limits_{k \in {B_{m}{(n)}}}{\frac{\alpha_{k}(n)}{\ln\; 2}\ln{{I_{N_{k}} + {{H_{m,k}(n)}{Q_{m,k}(n)}{H_{m,k}^{H}(n)}{R_{m,k}(n)}^{- 1}}}}}}}} + {\sum\limits_{m = 1}^{M}{\lambda_{m}\left\lbrack {P_{m,\max} - {\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {Q_{m,k}(n)} \right)}}}} \right\rbrack}} + {\sum\limits_{m = 1}^{M}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {{Q_{m,k}(n)}{\Delta_{m,k}(n)}} \right)}}}}}},$where λ=(λ₁, . . . , λ_(M))^(T) is the vector of non-negative Lagrangianmultipliers associated to the transmit power constraints and{Δ_(m,k)(n)} are the Lagrangian multipliers associated with the positivesemi-definite constraints on {Q_(m,k)(n)}

Also, define the following leakage matrix L_(m,k)(n)εC^(P×P)

${{L_{m,k}(n)} = {\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{\frac{\alpha_{u}(n)}{\ln(2)}{H_{m,u}^{H}(n)}\left( {{R_{j,u}(n)}^{- 1} - \left( {{R_{j,u}(n)} + {{H_{j,u}(n)}{Q_{j,u}(n)}{H_{j,u}^{H}(n)}}} \right)^{- 1}} \right){H_{m,u}(n)}}}}},$which accounts for the interference caused by base station m to otherco-channel terminals on tone n when serving user kεB_(m)(n). By settingthe gradient of Λ(Q,λ) with respect to Q_(m,k)(n) equal to zero, thefollowing set of equalities are obtained:L _(m,k)(n)+ln 2(λ_(m) I _(P)−Δ_(m,k)(n))=α_(k)(n)H _(m,k) ^(H)(n)(R_(m,k)(n)+H _(m,k)(n)Q _(m,k)(n)H _(m,k) ^(H)(n))⁻¹ H _(m,k)(n), kεB_(m)(n), m=1, . . . , M, n=1, . . . , NThe above equations, along with the sum-power constraints in the priorequation, the positivity constraints and the following complementaryslackness conditions:

${{\lambda_{m}\left\lbrack {P_{m,\max} - {\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {Q_{m,k}(n)} \right)}}}} \right\rbrack} = 0},{m = 1},\ldots\mspace{14mu},M,{{{tr}\left( {{Q_{m,k}(n)}{\Delta_{m,k}(n)}} \right)} = 0},{k \in {B_{m}(n)}},{m = 1},{\ldots\mspace{14mu} M},{n = 1},\ldots\mspace{14mu},N,$form a set of first-order necessary conditions for optimality, namelythe KKT conditions.

The precoder matrices can be obtained in a distributed way as follows.The system is initialized with a choice of {Q_(m,k)(n)} using which{L_(m,k)(n)} and {R_(m,k)(n)} are determined. The system implements twoloops. In the inner loop the system solves the following convexoptimization problem at each base-station m, for a given {L_(m,k)(n)}and {R_(m,k)(n)}:

$\begin{matrix}{{\underset{\{{Q_{m,k}{(n)}}\}}{argmax}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\left\lbrack {{{\alpha_{k}(n)}\log_{2}{{I_{N_{k}} + {{H_{m,k}(n)}{Q_{m,k}(n)}{H_{m,k}^{H}(n)}{R_{m,k}(n)}^{- 1}}}}} - {{tr}\left( {{L_{m,k}(n)}{Q_{m,k}(n)}} \right)}} \right\rbrack}}},\mspace{79mu}{{s.t.\mspace{11mu}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {Q_{m,k}(n)} \right)}}}} \leq P_{m,\max}},{{Q_{m,k}(n)} \in \overset{\_}{Q}},\mspace{79mu}{\forall{k \in {B_{m}(n)}}},{\forall n}} & \left\lbrack {{problem}\mspace{14mu} 2} \right\rbrack\end{matrix}$

Upon obtaining {Q_(m,k)(n)}, the matrices {R_(m,k)(n)} are updated andthe problem [problem 2] is again solved at each base station. The innerloop (for a fixed set of leakage matrices {L_(m,k)(n)}) is repeated tillconvergence or a maximum number of iteration is achieved. Then theleakage matrices {L_(m,k)(n)} are updated in the outer loop and theinner loop which involves updating {Q_(m,k)(n)} using [problem 2] and{R_(m,k)(n)} is initiated again. The outer loop itself is repeated tillconvergence or till a maximum number is reached.

In the following, this process is referred to as Iterative CoordinatedLinear Precoding (ICLP).

If the coordinated base stations share both user data and channel stateinformation, then they can act as a unique large array with distributedantenna elements and this case is the fully-connected cluster (FCC). AnFCC encompassing {tilde over (M)} base stations, each one equipped with{tilde over (P)} transmit antennas and where power pooling across basestations is possible, is mathematically equivalent to a PCC with M=1 andP={tilde over (M)}{tilde over (P)}. For such an FCC the optimal weightedsum rate is obtained using dirty-paper coding (DPC) and can bedetermined. Applying the ICLP algorithm (to the equivalent PCC model)the system can obtain a locally optimal solution.

For initialization, the system can extend the MaximumSignal-to-Leakage-plus-Noise Ratio (MSLNR) Beamforming to MaximumSignal-to-Leakage-plus-Noise Ratio (MSLNR) precoding. For any userkεB_(m)(n) the precoder or equivalently the covariance matrix can beobtained by maximizing

$\frac{\overset{\overset{{useful}\mspace{14mu}{signal}\mspace{14mu}{strength}\mspace{14mu}{for}\mspace{14mu}{user}\mspace{14mu} k\mspace{14mu}{on}\mspace{14mu}{slot}\mspace{14mu} n}{︷}}{{tr}\left( {{H_{m,k}(n)}{Q_{m,k}(n)}{H_{m,k}^{H}(n)}} \right)}}{{\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}\underset{\underset{{leakage}\mspace{14mu}{to}\mspace{14mu}{user}\mspace{14mu} u\mspace{14mu}{on}\mspace{14mu}{slot}\mspace{14mu} n}{︸}}{{tr}\left( {{H_{m,u}(n)}{Q_{m,k}(n)}{H_{m,u}^{H}(n)}} \right)}}} + 1}.$Obtaining Channel State Information:

The design procedures described in the above embodiments need channelestimates at each base station. In frequency division duplex (FDD)systems, each user can first obtain estimates of the (effective) channelmatrices using common pilots transmitted by its serving base station aswell as the dominant interfering base stations. The system can use acommon pilot which can be a reference signal whose form or structure isknown in advance to all users. For instance, user k served by basestation m can determine estimates of {H_(m,k)(n)} for a few resourceslots using common pilots transmitted by base station k on those slotsand then interpolate to obtain channel estimates for the remainingslots. Similarly, estimates of {H_(j,k)(n)} corresponding to aninterfering base station j can be obtained using common pilotstransmitted by base station j. The user should know the slots (orpositions in time and/or frequency) where these common pilots aretransmitted. For simplicity in channel estimation, the common pilotstransmitted by adjacent base stations as well as those transmitted viadifferent antennas of the same base station could be kept orthogonalusing non-intersecting positions or via (near) orthogonal spreadingsequences. The obtained estimates can then be quantized by the user andreported back to the serving base station via a uplink feedback channel.

On the other hand in time division duplex (TDD) systems channelreciprocity can be exploited. Based on common pilots transmitted by userk, its serving base station m can obtain estimates of {H_(m,k)(n)}.Using the same pilots an interfering base station j can obtain estimatesof {H_(j,k)(n)} which it can then quantize and send to base station m.

A key practical problem in coordinated beamforming and scheduling isthat of the channel estimates getting outdated. In particular, due tothe delay between the time when a set of channel estimates aredetermined (by users or base stations) and the time when some or all ofthe users are scheduled for data transmission based on those estimates,henceforth referred to as the scheduling delay, the channels seen by thescheduled users could have significantly changed and consequently therates assigned to them may not be achievable which can adversely affectthe system throughput.

One way to mitigate the aforementioned problem is as follows. Thescheduling delay is a function of the backhaul latency associated withthe inter-base station signaling as well as the schedulerimplementations at the base stations. The average value of thescheduling delay can be tracked by each base station. In FDD systems,each base station can then broadcast its measured value to all the usersthat can be served by it, once in every several scheduling intervals.Upon receiving this value each user can employ any suitable predictionalgorithm to obtain a predicted channel estimates. In particular,suppose the average scheduling delay value is D. Then, user k can firstdetermine estimates of {H_(m,k)(n)},{H_(j,k)(n)}_(j≠m) corresponding totime t and then obtain predicted estimates which are its estimates ofchannels for time t+D. The user can quantize and feedback thesepredicted channels. Moreover, the UE can monitor the quality of itsprediction over a time window. If it finds that the prediction qualityis poor due to its mobility or other factors, it can inform its servingbase station which can take appropriate action such as dropping it fromthe set of users that are served via coordinated beamforming. In TDDsystems the prediction can be done by the base station itself. Thepredicted estimates are then used in the beamforming or precoding designprocedure.

The above embodiments address inter-cell interference mitigation in aMISO or MIMO wireless cellular network with universal frequency reuse.Each base station is equipped with multiple transmit antennas andspace-division multiple-access (SDMA) is employed to serve multipleco-channel users. The set of beamforming vectors or precoding matricesacross a cluster of coordinated cells and resource slots are designed soas to maximize the (instantaneous) weighted system sum-rate subject toper-base-station power constraints.

In a MISO system each user is equipped with a single receive antenna andreceives a single data stream from its serving base station. AnIterative Coordinated Beam-Forming (ICBF) algorithm is provided todesign the transmit beamforming vectors. The ICBF algorithm can also beapplied in an equivalent MISO network model wherein each user isequipped with multiple receive antennas and performs rank-one receivebeamforming before detection. Numerical results indicate that theproposed ICBF technique significantly improves throughput with respectto uncoordinated beamforming strategies and achieves a large fraction ofcapacity. Also, power balancing across multiple resource slots is quitebeneficial in a PCC scenario, while providing only marginal gains in anFCC scenario.

In a MIMO system each user may have multiple receive antennas and mayreceive multiple spatially-multiplexed data streams. An IterativeCoordinated Linear Precoding (ICLP) Algorithm is provided to design thetransmit precoding matrices.

The proposed ICBF and ICLP algorithms admit a distributed implementationand are general enough to be used both in a partially-connected cluster(PCC), wherein the coordinated base stations do not share data symbolsand optimize their local beams based on the inter-cell channel gainsonly, and in a fully-connected cluster (FCC), wherein coordinated basestations act as a unique large array with distributed antenna elements.Distributed implementations with reduced signaling among base stationscan also be done, and the system works robustly even with imperfectchannel knowledge.

The above process enhances a multi-cell wireless network with universalfrequency reuse and treats the problem of co-channel interferencemitigation in the downlink channel. Assuming that each base stationserves multiple single-antenna mobiles via space-divisionmultiple-access (SDMA), the system jointly optimizes the linearbeam-vectors or precoding matrices across a set of coordinated cells andresource slots: the objective function to be maximized is the weightedsystem sum-rate subject to per-base-station power constraints. Afterderiving the general structure of the optimal beam-vectors, theiterative process solves the Karush-Kuhn-Tucker (KKT) conditions of thenon-convex primal problem. Various approaches can be used to choose theinitial beam-vectors, one of which maximizes thesignal-to-leakage-plus-noise ratio. Simulation results are discussed inthe incorporated by reference provisional application to assess theperformance of the process.

1. A method to optimize wireless communication for a plurality of mobilewireless devices, comprising: a. determining a structure for downlinkbeamforming vectors optimizing a weighted sum-rate across multipleorthogonal resource slots in the time, frequency or code domain; and b.distributing a non-convex optimization exploiting the determinedstructure over a plurality of base stations.
 2. The method of claim 1,wherein the non-convex optimization is responsive to one or more of: aset of channel estimates; user priorities; and maximum transmit poweravailable at each base-station.
 3. The method of claim 1, wherein thenon-convex optimization utilizes Karush-Kuhn-Tucker (KKT) equations. 4.The method of claim 1, wherein the distributing over the base stationsincludes providing a small number of channel estimates at each basestation and limited message passing among adjacent base stations.
 5. Themethod of claim 1, comprising obtaining channel estimates.
 6. The methodof claim 1, comprising obtaining predicted channel estimates responsiveto an average scheduling delay parameter.
 7. The method of claim 1,wherein the mobile wireless device comprises a multiple-inputsingle-output (MISO) system wherein each user is equipped with a singlereceive antenna and receives a single data stream from its serving basestation via space-division multiple-access (SDMA).
 8. The method ofclaim 1, comprising determining the optimal structure of the beamformingvectors optimizing the weighted sum-rate in a MISO system.
 9. The methodof claim 8, comprising determining a leakage matrix.
 10. The method ofclaim 9, wherein determining a leakage matrix L_(m,k)(n) comprises:${L_{m,k}(n)} = {{\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{{P_{j,u}(n)}{G_{m,u}(n)}}}} \in C^{P}}$where${{P_{j,u}(n)} = \frac{{\alpha_{u}(n)}S\; I\; N\;{R_{j,u}(n)}}{1 + {\sum\limits_{l = 1}^{M}{\sum\limits_{s \in {B_{l}{(n)}}}{{w_{l,s}^{H}(n)}{G_{l,u}(n)}{w_{l,s}(n)}}}}}};$B_(m)(n), B_(l)(n), and B_(j)(n) denote a set of users served by basestations m, l, and j, respectively, on slot n; w_(m,k)(n)εC^(P) andw_(l,s)(n)εC^(P) denote the P-dimensional beam vector used by the basestations m and l, respectively, to serve users kεB_(m)(n) andsεB_(m)(n), respectively, on slot n; P is the number of transmitantennas at each base station; W={w_(m,k)(n), kεB_(m)(n), m=1, . . . ,M, n=1, . . . N} indicates the collection of all beam vectors; N is thetotal number of orthogonal resource slots; M is the number ofcoordinated base stations; α_(k)(n)>0 is priority assigned by ascheduler to user k on slot n; SINR_(m,k)(n) and SINR_(j,u)(n) denotethe signal-to-interference-plus-noise ratio for users k and u,respectively, served by base stations m and j, respectively, on slot nand are given by the form of${S\; I\; N\;{R_{m,k}(n)}} = \frac{{w_{m,k}^{H}(n)}{G_{m,k}(n)}{w_{m,k}(n)}}{1 + {\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{{w_{j,u}^{H}(n)}{G_{j,k}(n)}{w_{j,u}(n)}}}}}$ with G_(m,k)(n)=h_(m,k)(n)h_(m,k) ^(H)(n)εC^(P×P) and G_(l,u)(n) andG_(j,k)(n) have the same form as G_(m,k)(n); h_(m,k)(n)εC^(P) is aP-dimensional channel vector between base station m and user k and slotn (which includes small-scale fading, large scale fading and pathattenuation) normalized by the standard deviation of the received noise.11. The method of claim 10, comprising determining${{i_{m,k}(n)} = {\sum\limits_{l = 1}^{M}{\sum\limits_{\underset{{({l,s})} \neq {({m,k})}}{s \in {B_{l}{(n)}}}}{{w_{l,s}^{H}(n)}{G_{l,k}(n)}{w_{l,s}(n)}}}}},{k \in {B_{m}(n)}},{m = 1},\ldots\mspace{14mu},M,{n = 1},{\ldots\mspace{14mu} N}$where i_(m,k)(n) stands for a scalar.
 12. The method of claim 8,comprising determining λ₁, . . . , λ_(M) and{B̂_(m, k)(n; λ_(m)), k ∈ B_(m)(n), n = 1, …  N}  $\mspace{79mu}{{{using}\mspace{14mu}{{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}}^{2}} = \frac{\left( {{{\alpha_{k}(n)}{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}} - {{\hat{i}}_{m,k}(n)} - 1} \right)^{+}}{{{{h_{m,k}^{H}(n)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}}\mspace{14mu}$$\mspace{79mu}{{{and}\mspace{14mu}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\underset{\underset{f_{m}{({n;\lambda_{m}})}}{︸}}{{{{{\hat{\beta}}_{m,k}\left( {n;\lambda_{m}} \right)}{{\hat{T}}_{m,k}^{\dagger}\left( {n;\lambda_{m}} \right)}{h_{m,k}(n)}}}^{2}}}}} = P_{m,\max}}$where {circumflex over (T)}_(m,k)(n;λ_(m))={circumflex over(L)}_(m,k)(n;λ_(m))+(λ_(m) ln 2)I_(p), (•)^(†)indicates thepseudo-inverse and x⁺=max{x,0}, {î_(m,k)(n),{circumflex over(L)}_(m,k)(n)} denotes most recent values of {i_(m,k)(n),L_(m,k)(n)},λ₁, . . . , λ_(M) are non-negative Lagrangian multipliers associated tothe transmit power constraints, α_(k)(n)>0 is priority assigned by ascheduler to user k on slot n, P_(m,max) is the maximum transmit powerof base station m, L_(m,k)(n) is a leakage matrix, h_(m,k)(n)εC^(P) is aP-dimensional channel vector between base station m and user k and slotn (which includes small-scale fading, large scale fading and pathattenuation) normalized by the standard deviation of the received noise,P is the number of transmit antennas at each base station, B_(m)(n)denotes a set of users served by base station m on slot n, and N is thetotal number of orthogonal resource slots, M is the number ofcoordinated base stations.
 13. The method of claim 12, comprisingupdating beam vectors asw _(m,k)(n)={circumflex over (β)}_(m,k)(n;λ _(m)){circumflex over (T)}_(m,k) ^(†)(n;λ _(m))h _(m,k)(n).
 14. The method of claim 8, comprisingchoosing an initial feasible set of beam vectors.
 15. The method ofclaim 14, wherein the initial feasible set is selected after splittingpower across the available slots.
 16. The method of claim 14, comprisingperforming channel-matched beamforming to select the initial feasibleset of beam vectors.
 17. The method of claim 14, comprising performingin-cell zero-forcing beamforming to select the initial feasible set ofbeam vectors.
 18. The method of claim 14, comprising performing maximumsignal-to-leakage-plus-noise ratio (MSLNR) beamforming to select theinitial feasible set of beam vectors.
 19. The method of claim 1,comprising providing an equivalent MISO system wherein each mobile userhas multiple-receive antennas and performs rank-one receive beamformingbefore detection.
 20. The method of claim 1, wherein the mobile wirelessdevice comprises a multiple-input multiple-output (MIMO) system andwherein each user is equipped with multiple receive antennas andreceives multiple data streams from a serving base station via linearprecoding.
 21. The method of claim 20, comprising determining a leakagematrix.
 22. The method of claim 20, comprising solving a followingconvex optimization problem at each base-station m, for a given{L_(m,k)(n)} and {R_(m,k)(n)}:${\underset{\{{Q_{m,k}{(n)}}\}}{argmax}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}\left\lbrack {{{\alpha_{k}(n)}\log_{2}{{I_{N_{k}} + {{H_{m,k}(n)}{Q_{m,k}(n)}{H_{m,k}^{H}(n)}{R_{m,k}(n)}^{- 1}}}}} - {{tr}\left( {{L_{m,k}(n)}{Q_{m,k}(n)}} \right)}} \right\rbrack}}},{{s.t.\mspace{11mu}{\sum\limits_{n = 1}^{N}{\sum\limits_{k \in {B_{m}{(n)}}}{{tr}\left( {Q_{m,k}(n)} \right)}}}} \leq P_{m,\max}},{{Q_{m,k}(n)} \in \overset{\_}{Q}},{\forall{k \in {B_{m}(n)}}},{\forall n}$where L_(m,k)(n) is a leakage matrix, R_(m,k)(n) is a covariance matrixof a co-channel interference plus noise seen by user kεB_(m)(n) on slotn, Q_(m,k)(n)=W_(m,k)(n)W_(m,k) ^(H)(n)εC^(P×P) is a covariance matrixto be optimized, W_(m,k)(n) indicates a precoding matrix used to serveuser kεB_(m)(n) on slot n by base station m; N is the total number oforthogonal resource slots; M is the number of coordinated base stations;P is the number of transmit antennas at each base station; B_(m)(n) is aset of users served by base station m on slot n; P_(m,max) is themaximum transmit power of base station m, and tr(•) denotes the traceoperation H_(m,k)(n)εC^(N) ^(R) ^(×P) is a channel matrix between basestation m and user k and slot n (which includes small-scale fading,large scale fading and path attenuation) normalized by the standarddeviation of the received noise.
 23. The method of claim 20, comprisingselecting an initial feasible set of positive semi-definite covariancematrices after splitting power across available slots.
 24. The method ofclaim 23, comprising performing Maximum Signal-to-Leakage-plus-NoiseRatio (MSLNR) precoding to select the initial feasible set of positivesemi-definite covariance matrices.
 25. The method of claim 1, comprisingdetermining an optimal structure of the linear precoders in a MIMOsystem.
 26. The method of claim 21, wherein determining a leakage matrixL_(m,k)(n) comprises determining:${{L_{m,k}(n)} = {\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{\frac{\alpha_{u}(n)}{\ln(2)}{H_{m,u}^{H}(n)}\left( {{R_{j,u}(n)}^{- 1} - \begin{pmatrix}{{R_{j,u}(n)} + {H_{j,u}(n)}} \\{{Q_{j,u}(n)}{H_{j,u}^{H}(n)}}\end{pmatrix}^{- 1}} \right){H_{m,u}(n)}}}}},\mspace{79mu}{where}$$\mspace{79mu}{{R_{m,k}(n)} = {{\sum\limits_{j = 1}^{M}{\sum\limits_{\underset{{({j,u})} \neq {({m,k})}}{u \in {B_{j}{(n)}}}}{{H_{j,k}(n)}{Q_{j,u}(n)}{H_{j,k}^{H}(n)}}}} + I_{N_{k}}}}$and R_(j,u)(n) has the same form as R_(m,k)(n); B_(m)(n) is a set ofusers served by base station m on slot n; N_(k) is the number of receiveantennas of user k, W_(m,k)(n)εC^(P×D) ^(m,k) ^((n)) andW_(j,u)(n)εC^(P×D) ^(j,u) ^((n)) indicate precoding matrices used toserve users kεB_(m)(n) and uεB_(j)(n), respectively, on slot n by basestations m and j, respectively, and D_(m,k)(n) and D_(j,u)(n) denote thenumber of data streams delivered to users kεB_(m)(n) and uεB_(j)(n) onslot n by base stations m and j, respectively; P is the number oftransmit antennas at each base station; Q={Q_(m,k)(n)ε Q, kεB_(m)(n),m=1, . . . , M, n=1, . . . N} indicates the collection of thepositive-semidefinite covariance matrices; N is the total number oforthogonal resource slots; M is the number of coordinated base stations;α_(k)(n)>0 is priority assigned by a scheduler to user k on slot n;H_(j,k)(n)εC^(N) ^(k) ^(×P) is a channel matrix between base station jand user k and slot n (which includes small-scale fading, large scalefading and path attenuation) normalized by the standard deviation of thereceived noise.
 27. A communication system, comprising: a. a pluralityof mobile devices that receive data made by linear precoding matriceshaving a predetermined structure for downlink precoding, thepredetermined structure being optimal with respect to a weighted sumrate in a multi-cell orthogonal frequency division multiple access(OFDMA) downlink; and b. a plurality of base stations communicating withthe mobile devices, all base stations performing a distributed anon-convex optimization exploiting the predetermined structure.